Abstract
Given two binary trees on N labeled leaves, the quartet distance between the trees is the number of disagreeing quartets. By permuting the leaves at random, the expected quartet distance between the two trees is \(\frac{2}{3}\left( {\begin{array}{c}N\\ 4\end{array}}\right) \). However, no strongly explicit construction reaching this bound asymptotically was known. We consider complete, balanced binary trees on \(N=2^n\) leaves, labeled by n bits long sequences. Ordering the leaves in one tree by the prefix order, and in the other tree by the suffix order, we show that the resulting quartet distance is \(\left( \frac{2}{3} + o(1)\right) \left( {\begin{array}{c}N\\ 4\end{array}}\right) \), and it always exceeds the \(\frac{2}{3}\left( {\begin{array}{c}N\\ 4\end{array}}\right) \) bound.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alon, N., Naves, H., Sudakov, B.: On the maximum quartet distance between phylogenetic tress. SIAM J. Discrete Math. 30(2), 718–735 (2016)
Alon, N., Snir, S., Yuster, R.: On the compatibility of quartet trees. SIAM J. Discrete Math. 28(3), 1493–1507 (2014)
Bandelt, H.J., Dress, A.: Reconstructing the shape of a tree from observed dissimilarity data. Adv. Appl. Math. 7(3), 309–343 (1986)
Ben-Dor, A., Chor, B., Graur, D., Ophir, R., Pelleg, D.: Constructing phylogenies from quartets: elucidation of eutherian superordinal relationships. J. Comput. Biol. 5(3), 377–390 (1998)
Berry, V., Jiang, T., Kearney, P., Li, M., Wareham, T.: Quartet cleaning: improved algorithms and simulations. In: Jaroslav Nešetřil (ed.) Algorithms — ESA’ 99, Lecture Notes in Comput. Sci., Vol. 1643, 313–324. Springer, Berlin (1999)
Brodal, G.S., Fagerberg, R., Pedersen, C.N.S.: Computing the quartet distance between evolutionary trees in time \(O(n\log n)\). Algorithmica 38(2), 377–395 (2004)
Estabrook, G., McMorris, F., Meacham, C.: Comparison of undirected phylogenetic trees based on subtrees of four evolutionary units. Syst. Biol. 34(2), 193–200 (1985)
Indyk, P., Ngo, H.Q., Rudra, A.: Efficiently decodable non-adaptive group testing. In: Charikar, M. (ed.) Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, 1126–1142. SIAM, Philadelphia, PA (2010)
Semple, C., Steel, M.: Phylogenetics. Oxford Lecture Series in Mathematics and Its Applications, Vol. 24. Oxford University Press, Oxford (2003)
Snir, S., Rao, S.: Quartets MaxCut: a divide and conquer quartets algorithm. IEEE/ACM Trans. Comput. Biol. Bioinf. 7(4), 714–718 (2010)
Steel, M.: The complexity of reconstructing trees from qualitative characters and subtrees. J. Classification 9(1), 91–116 (1992)
Strimmer, K., von Haeseler, A.: Quartet puzzling: a quartet maximum-likelihood method for reconstructing tree topologies. Mol. Biol. Evol. 13(7), 964–969 (1996)
Acknowledgements
Thanks to Stefan Grünewald for helpful discussions, which motivated this work. We would also like to thank Noga Alon and Gil Cohen for their help regarding what “explicit construction” exactly means. PLE was supported in part by the Hungarian NSF Grant K116769. Part of this work was done when PLE visited BC, supported by an exchange program of the Hungarian and Israeli Academies of Sciences. BC was supported by a Grant from the Blavatnik Computer Science Research Fund, and by the LTZI (Long Term Zero Income) fund of the ISF (Israeli Science Foundation).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Chor, B., Erdős, P.L. & Komornik, Y. A High Quartet Distance Construction. Ann. Comb. 23, 51–65 (2019). https://doi.org/10.1007/s00026-018-0411-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00026-018-0411-3